The Structure of a Viscoplastic Fluid Flow during Filling of a Circular Pipe/Plane Channel

Abstract

The viscoplastic fluid flow initiated in a circular pipe/plane channel during its filling in the gravity field at the flow rate specified at the inlet section is investigated. A mathematical formulation of the problem is stated based on complete equations of motion, the continuity equation, the natural boundary conditions on a free surface, and no-slip boundary condition on the solid wall. The rheological behavior of the medium is described by the Schwedoff–Bingham law, which presupposes the existence of quasi-solid motion zones (unyielded zones) in regions of low strain rates. The numerical solution of the problem is based on the finite-difference approach including the finite volume method and SIMPLE algorithm for calculating velocity and pressure fields at the internal nodes of a staggered grid. The method of invariants is used to satisfy the boundary conditions on the free surface. To provide a through computation of the flow with unyielded regions, regularization of the rheological equation is implemented. The behavior of a free boundary, flow structure, and flow characteristics as a function of the main parameters is investigated. It is found that in the course of time the initially flat free boundary acquires a stationary convex shape, which remains invariant while moving through the pipe/channel at the rate-average velocity. In the flow near and far away from the free boundary, one can distinguish fountain flow zones and one-dimensional flow regions, respectively. The typical flow structures with different numbers and various locations of unyielded regions in the flow are shown. The topograms of the above-mentioned flow structures as functions of the ratio of viscous and gravity forces and plasticity in the fluid flow are plotted. The stable and unstable behavior of the free boundary shape is shown to be related to the values of the constitutive parameters.